Combinations

Combinations: Learn

A combination is an arrangement in which order does not matter.
The notation for combinations is C(n,r) which is the number of
combinations of "n" things if only "r" are selected.

For example, in most card games, the order of the cards in your hand
does not matter, only which cards are in your hand.

Since order does not matter, that means there are fewer possibile
combinations compared to permutations. We can actually divide the
permutations by r! to find the number of combinations. The final
formula for combinations is:

C(n,r) =

n!

r!(n-r)!

= number of total possible combinations

For example, there is a card game where you must make the best 5-card
hand out of 7 available cards.

C(7,5) =

7!

5!(7-5)!

=

7*6*5*4*3*2*1

5*4*3*2*1*2*1

=

7*6*5*4*3*2*1

5*4*3*2*1*2*1

=

7*6

2*1

=

42

2

= 21 possibilities.

Notice how after writing out the factorial in the fraction, you can start
to reduce the fraction by canceling most of the factors. This makes
factorials the easy way to find how many possible permutations are available.